Towards Scalable Exoplanet Atmospheric Retrieval with Hierarchical Flow Matching Posterior Estimation

Abstract. Atmospheric retrieval is a fundamental tool for the characterization of exoplanetary atmospheres, enabling a deeper understanding of planetary formation, evolution, and habitability. Yet, traditional Bayesian retrieval methods remain computationally prohibitive at the scale demanded by upcoming surveys such as the ESA Ariel mission.  We present Hierarchical Flow Matching Posterior Estimation (HFMPE), a novel simulation-based inference framework for exoplanetary atmospheric retrieval, which efficiently provides high-quality posterior distributions using a hierarchical formulation. Benchmarked on the Ariel Data Challenge 2023 dataset against neural- and sampling-based competitors, HFMPE achieves superior or on-par retrieval performance in terms of regression errors, posterior calibration, uncertainty quantification, and coverage at lower computational cost, highlighting its potential as an efficient and scalable tool for next-generation atmospheric characterisation.

Introduction. Atmospheric retrieval is the primary method for inferring the physical and chemical properties of exoplanetary atmospheres. including molecular abundances, thermal profiles, and cloud or haze properties, from observed spectroscopic data [1]. This technique is therefore fundamental to characterizing exoplanetary atmospheres, and by extension, to advancing our understanding of planetary formation, evolution, and the habitability conditions of these distant worlds. The next generation of exoplanet surveys, including the ESA Ariel mission [2], will observe thousands of targets, demanding accurate posterior inference at unprecedented scale. However, traditional Bayesian approaches, such as Markov Chain Monte Carlo and Nested Sampling [3], while providing asymptotically exact uncertainty estimates, are inherently sequential and computationally prohibitive for datasets of this magnitude. 

Background. Recent simulation-based inference methods, such as Flow Matching Posterior Estimation (FMPE) [4], address this challenge by significantly accelerating retrieval while maintaining highly competitive posterior accuracy. Specifically, FMPE is a simulation-based inference technique built on Continuous Normalizing Flows [5], which trains a neural network to learn a time-dependent velocity field that transports samples from a simple prior distribution to the target posterior. At inference time, this transport is simulated by integrating an ordinary differential equation (ODE), yielding high-quality posterior samples in a fraction of the time required by classical methods [6, 7]. Despite this advantage, FMPE still demands a substantial number of neural function evaluations (NFEs) to achieve accurate integration, and reducing this cost is key to further improving its scalability for large-scale surveys.

Contribution. Inspired by Hierarchical Rectified Flow [8], we introduce Hierarchical Flow Matching Posterior Estimation (HFMPE), a novel retrieval framework that hierarchically couples multiple ODEs operating in different domains (location, velocity, acceleration, etc.). This formulation produces straighter and more efficient sampling trajectories, substantially reducing the computational burden during inference. The proposed framework naturally extends our prior work [7], which jointly exploits transmission spectra, per-channel instrumental uncertainties, and auxiliary planetary system parameters to estimate the full posterior distribution of atmospheric parameters.

Validation. We benchmark HFMPE on the Ariel Data Challenge (ADC) 2023 dataset [9] against both state-of-the-art neural and sampling-based baselines, including the FMPE baseline, Neural Posterior Estimation (NPE) with discrete normalizing flows [10], and Nested Sampling. We adopt a comprehensive evaluation protocol encompassing regression errors, posterior calibration, uncertainty quantification, and coverage. HFMPE outperforms NPE across most predictive metrics and achieves superior or on-par performance relative to Nested Sampling and FMPE with fewer neural function evaluations, demonstrating its effectiveness and efficiency as a retrieval tool for large-scale atmospheric characterization.

Conclusion.  In this talk, I will present HFMPE, a novel, scalable atmospheric retrieval framework, focusing on its motivation, mathematical background and experimental validation. We show that HFMPE converges towards state-of-the-art performance with fewer neural function evaluations on the ADC dataset, and demonstrates even more promising retrieval performance at comparable inference budgets. These results highlight HFMPE’s potential as an efficient and scalable solution for next-generation exoplanet atmospheric characterization, particularly in the context of large-scale survey missions such as Ariel.

The authors acknowledge financial contribution from the European Union - Next Generation EU RRF M4C2 1.1 PRIN MUR 2022 project 2022CERJ49 (ESPLORA) "Finanziato dall'Unione europea- Next Generation EU, Missione 4 Componente 2 CUP Master C53D23001060006, CUP I53D23000660006".

References. [1] Madhusudhan, N. 2019, Annual Review of Astronomy and Astrophysics, 57(1), 617–663; [2] Gargaud, M. et al. (eds.) 2023, Atmospheric Remote-Sensing Infrared Exoplanet Large-Survey, Springer, Berlin, Heidelberg, pp. 275–275; [3] Feroz, F. et al. 2019, The Open Journal of Astrophysics, 2; [4] Wildberger, J.B. et al. 2023, Thirty-seventh Conference on Neural Information Processing Systems; [5] Chen, R.T.Q. et al. 2018, Advances in Neural Information Processing Systems, 31; [6] Gebhard, T.D. et al. 2025, Astronomy & Astrophysics, 693, 42; [7] Giordano Orsini, M. et al. 2025, IEEE Access, 1–1; [8] Zhang, Y. et al. 2025, Thirteenth International Conference on Learning Representations; [9] Changeat, Q. & Yip, K.H. 2023, RAS Techniques and Instruments, 2(1), 45–61; [10] Papamakarios, G. et al. 2021, J. Mach. Learn. Res., 22(1)